Question about RG and scaling in qft

In summary, In critical phenomena, the block size can be enlarged by using Kadanoff transformation and the new Hamiltonian can be scaled to recover the original block size. In QFT, integrating out high momenta produces the effective Lagrangian, where the parameters y depend on the scaling factor b. To get a new effective Lagrangian with a different cut-off, we can scale the parameters k' and x' and the new parameters y' will also depend on b. The beta functions are defined as dy/db, as they measure how the parameters change with the cut-off.
  • #1
beacon
8
0
In critical phenomena, we can enlarge the block size(momenta fluctuation) by Kadanoff transformation, say
[tex]k \rightarrow bk (b<=1) [/tex], and scale the new Hamiltonian by [tex]k' = k/b, x'=bx[/tex] to recover to the original block size.
In QFT, similarly integrating out the high momenta produces the effective Langrangian,

[tex]\int_{k<=b\Lambda} [D\phi] exp(iS_{eff}) = \int_{b\Lambda <k < \Lambda} [D\phi] exp(iS)[/tex].

The parameters [tex]y[/tex] in the effective langrangian [tex]S_{eff}[/tex] should depend on [tex]b[/tex]. We can also do a scaling [tex]k' = k/b, x'=bx[/tex] in [tex]S_{eff}[/tex] to get [tex]S'_{eff}[/tex] whose path integral is now [tex]\int_{k' <= \Lambda}[/tex]. The parameters [tex]y'[/tex] also depend on [tex]b[/tex]. My puzzle is that which are the so-called beta fuctions, [tex]dy \over db[/tex] or [tex]dy' \over db[/tex]
 
Last edited:
Physics news on Phys.org
  • #2
The beta functions are usually defined as dy/db. This is because the beta functions measure how the parameters of the theory change as the cut-off is varied. When we scale the parameters k' and x', the new parameters y' do not depend on the cut-off anymore, so they are not relevant for the beta functions.
 
  • #3


In QFT, the concept of renormalization group (RG) plays a crucial role in understanding the behavior of a theory at different length scales. The idea is to integrate out high momenta modes and study how the theory changes as we vary the energy scale or the size of the block. This is analogous to the Kadanoff transformation in critical phenomena, where we enlarge the block size by a factor of b and then rescale the new Hamiltonian and coordinates to recover the original block size.

In the context of QFT, integrating out high momenta modes leads to an effective Lagrangian, which contains all the relevant information about the theory at a given energy scale. This effective Lagrangian depends on a parameter b, which is related to the scale at which we are studying the theory. As you correctly pointed out, we can also perform a scaling transformation on the effective Lagrangian to get a new effective Lagrangian, which corresponds to studying the theory at a different energy scale. This new effective Lagrangian will have different parameters, denoted by y', which also depend on b.

Now, the beta function is defined as the rate of change of a parameter with respect to the scale b. In other words, it tells us how a parameter changes as we study the theory at different energy scales. In the context of your question, the beta function would be given by dy/db or dy'/db, depending on which effective Lagrangian we are considering. Both of these beta functions are important in understanding the behavior of the theory at different scales.

To summarize, the beta function is a crucial tool in studying the behavior of a theory under scale transformations, and both dy/db and dy'/db are important in understanding the behavior of the theory at different energy scales.
 

1. What is RG (Renormalization Group) in QFT (Quantum Field Theory)?

RG in QFT is a mathematical tool used to study the behavior of quantum field theories at different energy scales. It helps us understand how the properties of a system change as we zoom in or out on it. This is important because it allows us to make predictions about the behavior of a system at different energy scales without needing to know all the details of the theory.

2. How does RG help with scaling in QFT?

RG helps with scaling in QFT by allowing us to track how the properties of a system change as we change the energy scale. This is important because many quantum field theories exhibit scaling behavior, where the properties of the system at different energy scales are related in a specific way. RG helps us understand and predict this behavior.

3. What is the relationship between RG and critical points in QFT?

RG and critical points in QFT are closely related. Critical points are points in the phase diagram of a system where the system undergoes a phase transition. RG helps us understand and predict the behavior of a system near critical points, and can even determine the nature of the phase transition.

4. How is RG used in practical applications of QFT?

RG is used in practical applications of QFT in various ways. For example, it is used to study phase transitions in condensed matter systems, to understand the behavior of particles in high energy physics experiments, and to make predictions about the behavior of systems at extreme temperatures or energy scales.

5. Are there any limitations or challenges when using RG in QFT?

Yes, there are some limitations and challenges when using RG in QFT. One challenge is that RG calculations can become very complicated and difficult to solve for more complex systems. Additionally, RG is not always applicable to all quantum field theories, and it may not accurately describe certain behaviors, such as non-equilibrium processes. However, it is still a powerful tool for understanding and predicting the behavior of many quantum field theories.

Similar threads

  • High Energy, Nuclear, Particle Physics
Replies
2
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
9
Views
2K
  • Quantum Physics
Replies
9
Views
6K
  • Math Proof Training and Practice
6
Replies
175
Views
20K
  • Beyond the Standard Models
Replies
7
Views
3K
  • MATLAB, Maple, Mathematica, LaTeX
Replies
3
Views
223
  • MATLAB, Maple, Mathematica, LaTeX
Replies
4
Views
3K
Back
Top